Best Known (172−75, 172, s)-Nets in Base 2
(172−75, 172, 54)-Net over F2 — Constructive and digital
Digital (97, 172, 54)-net over F2, using
- t-expansion [i] based on digital (95, 172, 54)-net over F2, using
- net from sequence [i] based on digital (95, 53)-sequence over F2, using
- Niederreiter–Xing sequence construction III based on the algebraic function field F/F2 with g(F) = 69, N(F) = 48, 1 place with degree 2, and 5 places with degree 6 [i] based on function field F/F2 with g(F) = 69 and N(F) ≥ 48, using an explicitly constructive algebraic function field [i]
- net from sequence [i] based on digital (95, 53)-sequence over F2, using
(172−75, 172, 67)-Net over F2 — Digital
Digital (97, 172, 67)-net over F2, using
(172−75, 172, 293)-Net in Base 2 — Upper bound on s
There is no (97, 172, 294)-net in base 2, because
- 1 times m-reduction [i] would yield (97, 171, 294)-net in base 2, but
- extracting embedded orthogonal array [i] would yield OA(2171, 294, S2, 74), but
- 5 times code embedding in larger space [i] would yield OA(2176, 299, S2, 74), but
- adding a parity check bit [i] would yield OA(2177, 300, S2, 75), but
- the linear programming bound shows that M ≥ 3 635392 612138 849947 593748 250106 276323 244402 593086 652587 881083 736015 264809 747844 602393 772282 952895 309564 287520 927482 512784 061552 651238 260008 282439 847339 186197 936622 918149 603328 / 16 596489 288891 597431 806767 668273 879726 323125 162254 820959 724017 639569 206174 043207 555202 380507 742012 504078 557367 932811 307081 > 2177 [i]
- adding a parity check bit [i] would yield OA(2177, 300, S2, 75), but
- 5 times code embedding in larger space [i] would yield OA(2176, 299, S2, 74), but
- extracting embedded orthogonal array [i] would yield OA(2171, 294, S2, 74), but